Finding the homomorphisms of these groups There are 3 questions that I've stuck.
Here are the questions.
Find all the group homomorphisms.
$(1)$ $\phi :  (Z,+) \times (Z,+) \to (2Z,+)$
$(2)$ $\phi :  (2Z,+) \to  (Z,+) \times (Z,+)$
$(3)$ $\phi : (Z,+)\to (S_3, \circ)$
I've found the homomorphisms by observing the generator is where to go by using the order of the groups.
For example of (1)[my trial]
$Z \times Z = \langle(1,0) ,(0,1)\rangle$
All we have to do is just finding the image of $\phi(1,0)$ and $\phi(0,1)$
But, $Z \times Z$ has a infinite order, So I couldn't find the image of the  $\phi(1,0)$ and $\phi(0,1)$.
Are there any effetive way to find the homomorphism when the infinite group case??
 A: The idea is simple : suppose you have a homomorphism of groups $\psi : G \to H$. For $a \in G$, if $\psi(a) = b \in H$ then $\psi(a^n) = b^n$ for every $n$. 
Now, if $a^n = e$ for some finite $n$, then $b^n = e$, so there is a restriction on $b$. However, $a^n \neq e$ for any finite $n$ (as is the case with $(0,1)$) then this restriction does not come into play at all! 
Which means, that $\phi(1,0)$ and $\phi(0,1)$ can both take any value from $(2\mathbb Z,+)$ (for the first example).
In other words, the set of all homomorphisms in the case $(1)$ is in $1-1$ correspondence between the elements of $2 \mathbb Z \times 2 \mathbb Z$. 
For example, take the element $(68,-74) \in 2\mathbb Z \times 2\Bbb Z$. Then, this corresponds to the homomorphism $\phi_{68,-74}$ for which $\phi(1,0) = 68$ and $\phi(0,1) = -74$. Based on this correspondence, it is also easy to find which homomorphisms are injective/surjective etc.

Something similar holds for the second question : the element $(2,0)$ which generates $2\mathbb Z$ can be mapped to any element of $\mathbb Z \times \mathbb Z$ because it has infinite order. One can now describe all the group homomorphisms.

Similarly in the third question, $\phi(1)$ can be any element of $S_3$! Observe however, that $S_3$ is finite, so all homomorphisms will have non-trivial kernel, which is how it differs from the previous examples. Otherwise, the idea is the same : the order of the input dictates the order of the output, but if the input order is infinite then the output can be freely chosen. 
Note : if you don't know what the kernel means, then no worries ; it is not important for the question itself, and classifies as additional information.
A: You just need to see where the generators go. (In fact they can go to everywhere in those problems, since all of them have infinite order)
